A rectangle is twice as long as it is broad. Find the rate of change of the perimeter when
the breadth of the rectangle is 1m and its area is changing at the rate of 18cm2
/s.
thanks
length=2*width
area=length*width= width*2width=2 width^2
width= sqrt (area/2)
Perimeter= 2 width + 2 length= 2width+4width=6*width
perimeter=6 sqrt(area/2)
dP/dt=6 d(sqrt(area/2))/dt
take the derivative, and you have it.
A rectangle is twice as long as it is broad. Find the rate of change of the perimeter when the breadth of the rectangle is 1m and its area is changing at the rate of 18cm2 /s.
Why did the rectangle go to the gym? Because it wanted to work on its shape!
Alright, let's solve this rectangle puzzle. The given information tells us that the breadth of the rectangle is 1m, and its area is changing at a rate of 18cm^2/s.
To find the rate of change of the perimeter when the breadth is 1m, we need to express the perimeter in terms of the breadth, and then differentiate it with respect to time.
Let's assume the breadth of the rectangle is 'b' meters, which means the length will be 2b meters because the rectangle is twice as long as it is broad.
The area of the rectangle is given by A = b * (2b) = 2b^2
Since we know that the area is changing at a rate of 18cm^2/s, which is 0.18m^2/s, we can differentiate the area expression with respect to time:
dA/dt = 4b * db/dt = 0.18
Now, we need to find db/dt, which represents the rate of change of the breadth. From the equation above, we have:
4b * db/dt = 0.18
db/dt = 0.18 / (4b)
For the given breadth of 1m, the rate of change of the breadth is:
db/dt = 0.18 / (4 * 1) = 0.045 m/s
To find the rate of change of the perimeter, we need to differentiate the perimeter expression with respect to the breadth and then multiply it by db/dt:
P = 2 * (b + 2b) = 6b
dP/db = 6
Finally, we can find the rate of change of the perimeter when the breadth is 1m:
dP/dt = (dP/db) * (db/dt) = 6 * 0.045 = 0.27 m/s
So, the rate of change of the perimeter when the breadth is 1m and the area is changing at a rate of 18cm^2/s is 0.27 m/s.
Hope you enjoyed this rectangle adventure!
To find the rate of change of the perimeter when the breadth of the rectangle is 1m and its area is changing at the rate of 18cm^2/s, we need to use the given information to determine the relationship between the perimeter, breadth, and area of the rectangle.
Let's denote the length of the rectangle as L and the breadth as B. We are given that the rectangle is twice as long as it is broad, so we can write L = 2B.
The area of the rectangle is given by A = L * B, and we know that the area is changing at a rate of 18cm^2/s. We can represent this as dA/dt = 18cm^2/s, where dt represents the change in time.
Differentiating both sides of the equation A = L * B with respect to time (t), we get:
dA/dt = (dL/dt) * B + L * (dB/dt)
Now, since dL/dt = 0 (since the length of the rectangle is not changing), we can simplify the equation to:
dA/dt = L * (dB/dt)
Substituting the value of L from the given relationship L = 2B, we get:
dA/dt = 2B * (dB/dt)
Now, we need to find the rate of change of the perimeter when the breadth of the rectangle is 1m. The perimeter of a rectangle is given by P = 2L + 2B. Substituting the value of L = 2B, we get:
P = 2(2B) + 2B
P = 4B + 2B
P = 6B
Differentiating both sides of the equation P = 6B with respect to B, we get:
dP/dB = 6
Now, we have the expressions for dA/dt and dB/dt in terms of B. We can use these to find the rate of change of the perimeter when the breadth of the rectangle is 1m.
Substituting the value of dA/dt = 18cm^2/s and B = 1m into the equation:
18cm^2/s = 2(1m) * (dB/dt)
Simplifying this equation, we get:
18cm^2/s = 2 * dB/dt
Dividing both sides of the equation by 2, we get:
9cm^2/s = dB/dt
Finally, we can substitute this value of dB/dt into the expression we found earlier for dP/dB:
dP/dB = 6
Since dB/dt = 9cm^2/s, we can multiply this value with dP/dB to get the rate of change of the perimeter:
dP/dt = dP/dB * dB/dt
dP/dt = 6 * 9cm^2/s
dP/dt = 54cm^2/s
Therefore, the rate of change of the perimeter when the breadth of the rectangle is 1m and its area is changing at the rate of 18cm^2/s is 54cm^2/s.